Frequency spectrum of the geomagnetic field harmonic coefficients from dynamo simulations

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Frequency spectrum of the geomagnetic field harmonic coefficients from dynamo simulations

Claire Bouligand, Nicolas Gillet, Dominique Jault, Nathanaël Schaeffer, Alexandre Fournier, Julien Aubert

To cite this version:

Claire Bouligand, Nicolas Gillet, Dominique Jault, Nathanaël Schaeffer, Alexandre Fournier, et al.. Frequency spectrum of the geomagnetic field harmonic coefficients from dynamo simulations.

Geophysical Journal International, Oxford University Press (OUP), 2016, 207, pp.1142 - 1157.

�10.1093/gji/ggw326�. �hal-01451189�

Frequency spectrum of the geomagnetic field harmonic

1

coefficients from dynamo simulations

2

C. Bouligand ¹ , N. Gillet ¹ , D. Jault ¹ , N. Schaeffer ¹ , A. Fournier ² , J. Aubert ²

1 Univ. Grenoble Alpes, CNRS, ISTerre, F-38000 Grenoble, France.

2 Institut de Physique du Globe de Paris, Sorbonne Paris Cit´e, Univ. Paris Diderot, CNRS, 1 rue Jussieu, F-75005 Paris, France.

3

4

SUMMARY

5

6

The construction of geomagnetic, archeomagnetic or paleomagnetic field models requires

7

some prior knowledge about the actual field, which can be gathered from the statistical

8

properties of the field over a variety of length-scales and time-scales. However, available

9

geomagnetic data on centennial to millennial periods are too sparse to infer directly these

10

statistical properties. We thus use high-resolution numerical simulations of the geody-

11

namo to test a method for estimating the temporal power spectra (or equivalently the auto-

12

covariance functions) of the individual Gauss coefficients that describe the geomagnetic

13

field outside the Earth’s fluid outer core. Based on the spectral analysis of our simulations,

14

we argue that a prior for the observational geomagnetic field over decennial to millennial

15

periods can be constructed from the statistics of the field during the short satellite era. The

16

method rests on the assumption that time series of spherical harmonic coefficients can be

17

considered as realisations of stationary and differentiable stochastic processes, namely or-

18

der 2 autoregressive (AR2) processes. In the framework of these processes, the statistics

19

of Gauss coefficients are well constrained by their variance and one or two time-scales.

20

We find that the time spectra in the dynamo simulations of all Gauss coefficients but the

21

axial dipole are well approximated by the spectra of AR2 processes characterized by only

22

one timescale. The process parameters can simply be deduced from instantaneous esti-

23

mates of the spatial power spectra of the magnetic field and of its first time derivative.

24

Some deviations of the Gauss coefficients statistics from this minimal model are also dis-

25

cussed. Characterizing the axial dipole clearly requires a more sophisticated AR2 process,

26

with a second distinct time-scale.

27

Key words: Dynamo: theories and simulations; Magnetic field; Rapid time variations;

28

Probability distributions; Time-series analysis; Inverse theory.

29

1 INTRODUCTION

30

The construction of global field models or of regional master-curves from geomagnetic records has

31

required the use of spatial and temporal regularizations (e.g., Jackson et al. 2000; Korte et al. 2009;

32

Th´ebault and Gallet 2010). Searching for models as smooth as possible (e.g., Constable and Parker

33

1988a) allows to retrieve the features that are reliably constrained by the data, but does not give access

34

to uncertainties on model coefficients. To address this issue, geomagnetic models have been produced

35

using prior information in the form of covariance matrices for the model parameters. These matrices

36

have been built using either some knowledge of the temporal variability of the present geomagnetic

37

field, which we will further discuss here, or spatial cross-covariances deduced from geodynamo simu-

38

lations (e.g., Fournier et al. 2013, 2015). Such prior information is particularly useful when modeling

39

the Earth’s magnetic field on historical and archeological time-scales, for which the data distribution

40

is sparse in both space and time, and is characterized by large measurements (and sometimes dating)

41

errors. Finally, prior information in the form of covariance matrices is a prerequisite for data assimila-

42

tion. For instance, knowledge of the analysis covariance matrix in sequential assimilation is necessary

43

to forecast future trajectories of the geomagnetic field (e.g., Gillet et al. 2015; Aubert 2015).

44

In the probabilistic framework of assimilation algorithms, geomagnetic spherical harmonic coef-

45

ficients are assumed to result from Gaussian processes. These are stationary stochastic processes fully

46

specified by their means and auto-covariance functions (MacKay 1998). As a matter of fact, the auto-

47

covariance function of any stationary stochastic process can be deduced from its frequency spectrum.

48

Analyses of geomagnetic records suggest that their power spectrum P behaves as P (f ) ∝ f ^−s in some

49

ranges of frequency f, with s the spectral index (e.g., Constable and Johnson 2005; Panovska et al.

50

2013). This defines scale invariance. The index value is related to the underlying physical processes

51

and to the statistical properties of the time-series.

52

Studies of the Earth dipole moment (Constable and Johnson 2005) suggest a flat energy density

53

spectrum (s = 0) for the longest time-scales (1 Myr or more). This spectrum steepens towards higher

54

frequencies, with a spectral index s ' 2 at millennial to centennial time-scales (Panovska et al. 2013)

55

and s ' 4 from centennial to inter-annual time-scales (De Santis et al. 2003). Considering the un-

56

signed dipole moment for the past 2 Myr, Brendel et al. (2007) and Buffett et al. (2013) found that

57

its spectrum, over millennial periods, has also a spectral index of 2, and made the analogy with spec-

58

tra from realisations of autoregressive stochastic processes of order one (AR1). These processes have

59

continuous but non differentiable samples. They are also known as Ornstein-Uhlenbeck processes and

60

are solutions of a Langevin-type equation (Gardiner 1985). Buffett et al. (2013) argued that the charac-

61

teristic time-scale of the deterministic part of the stochastic process that they constructed from dipole

62

series is set by the dipole decay time t _d . In this framework, this time is related to the escape time for

63

bistable systems that they also connect to the rate of magnetic reversals. Buffett (2015) also studied

64

the relation of this time to the duration of polarity transitions.

65

Although the axial dipole field has been the focus of many studies, the non-dipolar field is much

66

less documented. On time-scales shorter than a few centuries, order 2 autoregressive (AR2) stochastic

67

processes, whose samples are differentiable, have been introduced to define prior information about

68

the auto-covariance function of the Gauss coefficients when building global magnetic field models

69

over the observatory era (Gillet et al. 2013) and regional models over archeological periods (Hel-

70

lio et al. 2014). They are indeed consistent with a spectral index s = 4 at decadal periods. Gillet

71

et al. (2013) characterized the appropriate AR2 stochastic process from the variance and the secular

72

variation times of the spherical harmonic coefficients. They calculated these two quantities from the

73

geomagnetic spatial power spectrum of the geomagnetic field (Lowes 1974) and of its time-derivative

74

(the secular variation) obtained from a field model of the well documented satellite era. Considering

75

geomagnetic series as sample functions of stochastic processes with power spectrum P (f ) ∝ f ⁻⁴

76

gives an explanation to the occurrence of geomagnetic jerks, which are defined as abrupt changes in

77

the geomagnetic field second time derivative (Mandea et al. 2010).

78

Constructing field models from realisations of AR2 processes yields time series very similar to

79

observatory series (Brown 2015). However, the hypothesis that Gauss coefficients can be described

80

in terms of AR2 stochastic processes is not easily tested using geomagnetic observations because

81

we lack highly accurate, dense coverage data over a long enough time window. In particular, the

82

satellite era is too short in comparison with the decadal to centennial correlation times involved in

83

the evolution of the geomagnetic field. For this reason, it may be helpful to investigate the statistics

84

of individual coefficient series from numerical simulations of the geodynamo. Although calculated

85

for dimensionless numbers far from Earth-like parameters, numerical simulations provide us with

86

time series of Gauss coefficients that may be used to test assumptions about the statistics of the field

87

coefficients (Kuipers et al. 2009; Tanriverdi and Tilgner 2011; Meduri and Wicht 2016). A major issue

88

is the rescaling of the time axis (Lhuillier et al. 2011b; Christensen et al. 2012). Buffett et al. (2014)

89

and Buffett and Matsui (2015) have just achieved a comparison between the frequency spectrum of

90

the dipole term obtained from a numerical simulation and the theoretical spectrum expected for a

91

stochastic process. In both numerical and theoretical spectra, they distinguished three domains of

92

increasing frequencies for which the spectral index is, as described above for the observed field, s = 0,

93

s = 2 and s = 4. Then, they documented the transitions between the three frequency ranges, and

94

proposed a phenomenological interpretation of the two cut-off times: they suggest that they are related

95

to the dipole decay time t d and to the lifetime of convective eddies in the fluid core. Attributing the

96

different times to specific underlying mechanisms in the geodynamo models may help to compare

97

simulations and observations and to overcome the limitations of the numerical models.

98

Instead of focusing our analysis on the dipole field, we apply here stochastic modeling to spherical

99

harmonics of higher degree. We use high-resolution numerical simulations to test a simple recipe for

100

the auto-covariance function of the geomagnetic coefficients based on instantaneous models of the

101

field and its time variation. We find that the AR2 stochastic processes recently used as prior by Gillet

102

et al. (2013) and Hellio et al. (2014) do provide an approximation of the temporal power spectra for

103

individual Gauss coefficients in the numerical simulations. Based on these results, we argue that up to

104

millennial periods the auto-covariance function of Gauss coefficients of the actual geomagnetic field

105

can be described with only two parameters (or three for the axial dipole).

106

The manuscript is organized as follows. In section 2, we give an overview of stochastic pro-

107

cesses that we consider in this study to model the time evolution of geomagnetic Gauss coefficients.

108

In section 3, we first give the main characteristics of the three different numerical dynamo simulations

109

analysed throughout this study, before we describe the statistics (variance, correlation time and spec-

110

tra) of the generated Gauss coefficients. Next, we compare the frequency spectra of non-dipole Gauss

111

coefficients in our dynamo simulations with spectra predicted from the assumption that they are reali-

112

sations of order 2 stochastic processes with a single characteristic time-scale. Finally in section 4 we

113

describe possible deviations from spherical symmetry, and discuss the specific behavior of the axial

114

dipole at millennial and longer periods. Those considerations lead us to speculate about the possible

115

mechanisms underlying the time-scales of the stochastic processes that we have considered. We finally

116

discuss consequences for uncertainty estimates in field modeling.

117

2 STOCHASTIC MODELS FOR THE TIME EVOLUTION OF GAUSS COEFFICIENTS

118

As stated by the Wiener-Khinchin theorem (Van Kampen 2007), a stationary stochastic process x

119

of time t can be characterized either by its power spectrum P (f ) or by its auto-covariance function

120

C(τ ) = E(x(t)x(t + τ )), where E(. . .) stands for the statistical expectation. Those two quantities are

121

related through

122

P(f) = Z ∞

−∞ C(τ )e ^{−i2πf τ} dτ . (1)

123

We make below a connection between the stochastic processes that we use in this study and the pro-

124

cesses that have been previously employed to model the evolution of the geocentric axial dipole.

125

2.1 A three-parameter AR2 process for the axial dipole

126

Transition between power laws P(f) ∝ f ⁻⁴ , f ⁻² , and f ⁰ at respectively high, intermediate and low

127

frequencies have been documented for the Earth magnetic field (e.g., Constable and Johnson 2005;

128

Ziegler et al. 2011) as well as for dynamo numerical simulations (Olson et al. 2012; Davies and

129

Constable 2014; Buffett and Matsui 2015). Based on these observations, Hellio (2015) and Buffett

130

and Matsui (2015) introduced specific stochastic processes for modeling the time changes of the axial

131

dipole. Their two approaches are compared below.

132

In the following, we assume that the axial dipole coefficient samples a stochastic process x(t), of

133

non-zero average x ¯ = E(x), i.e. we consider a period of constant (normal or inverse) polarity. We

134

discuss the fluctuations y(t) = x(t) − x ¯ about this average. Hellio et al. (2014) proposed that y is a

135

realisation of an AR2 stochastic process, namely is solution of a differential equation of the form

136

d ² y

dt ² + 2χ dy

dt + ω ² y = ζ(t) , (2)

137

where ζ(t) is a white noise process, and the frequencies ω and χ are positive. The latter two conditions

138

ensure that the process is stationary. For χ > ω, the frequency spectrum exhibits f ⁻⁴ , f ⁻² and f ⁰

139

dependence at respectively high, intermediate and low frequencies. It can be expressed as (e.g. Yaglom

140

2004)

141

P(f) = 4χω ² σ ²

ω ² − (2πf) ^{2 2} + (4πχf ) ²

, (3)

142

where σ ² = E y ² . It is thus constrained by three quantities: the process variance σ ² , and the param-

143

eters χ and ω. The auto-covariance function is given by

144

C(τ ) = σ ² 2ξ

(χ + ξ)e ^{−(χ−ξ)|τ|} − (χ − ξ)e ^{−(χ+ξ)|τ|} , (4)

145

with ξ ² = χ ² − ω ² . The time ω ⁻¹ can be obtained as the square root of the ratio between the variance

146

of y and of its time derivative (Hellio 2015, p.50). Indeed, the auto-covariance function C is twice

147

differentiable at τ = 0, with

148

C ⁰⁰ (0) = d ² dτ ² C(τ )

_τ=0

= − σ ² ω ² , (5)

149

and we have also (Hulot and Le Mou¨el 1994):

150

C ⁰⁰ (0) = − E

" dy dt (t)

2 #

. (6)

151

Buffett and Matsui (2015) model instead the evolution of x(t) using the stochastic equation

152

dx

dt = v(x) + ^q D(x)Γ(t) , (7)

153

where Γ(t) is a red noise characterized by a Laplacian auto-covariance function, v(x) is a drift term

154

describing the slow evolution of the axial dipole moment, and D(x) defines the amplitude of random

155

fluctuations. Following Buffett et al. (2013, 2014) and Buffett and Matsui (2015), the latter two terms

156

may be approximated by v(x) ' − (x − x)/τ ¯ _s = − y/τ _s and D(x) ' D, yielding a stochastic equation

157

of the form

158

dy dt + y

τ _s = (t) , (8)

159

with (t) = √

DΓ(t). Since (t) is a Laplacian correlated noise, its evolution can be modeled by an

160

order one stochastic equation of the form (e.g., Jazwinski 2007)

161

d dt +

τ _f = ζ(t) , (9)

162

with ζ (t) a white noise process.

163

Combining equations (8) and (9) leads to an equation of the form

164

d ² y dt ² + 1

τ _s + 1 τ _f

! dy dt + y

τ _s τ _f = ζ (t) . (10)

165

With 2χ = 1/τ s + 1/τ _f and ω ² = 1/(τ s τ _f ), equation (10) defines an AR2 stochastic process similar

166

to that defined through equation (2). Adopting τ f < τ s , we obtain

167



 

 

τ s = χ + ξ ω ² τ f = χ − ξ

ω ²

. (11)

For τ f τ s , we deduce from (3) and (11) that the transition period between domains of the power

168

spectrum presenting 2 and 4 (resp. 0 and 2) spectral indices is 2πτ _f (resp. 2πτ _s ).

169

Hellio (2015) and Buffett et al. (2013) are therefore using similar stochastic models for the axial

170

dipole. Note however that the latter implicitly states the condition ξ real and χ ≥ ω – see equation

171

(11). Equation (2) is thus more general, and allows a wider range of behaviors.

172

2.2 A two-parameter AR2 process for non dipole coefficients

173

For an AR2 process with χ = ω (i. e. τ _f = τ _s ), the frequency spectrum of the process defined from

174

(2) is given by

175

P(f) = 4ω ³ σ ²

[ω ² + (2πf ) ² ] ² . (12)

176

This power spectrum is flat (spectral index s = 0) at low frequencies and behaves as f ⁻⁴ for f

177

ω/(2π). It does not display a power law f ⁻² at intermediate frequencies. The auto-covariance function

178

of the process is given by

179

C(τ ) = σ ² (1 + ω | τ | ) e ^−ω|τ| . (13)

180

This particular autoregressive process of order 2 depends only on two parameters, the variance σ ² and

181

the characteristic time-scale ω ⁻¹ . It was used by Gillet et al. (2013), Hellio et al. (2014), and Hellio

182

(2015) to define prior information on Gauss coefficients for the computation of global archeomagnetic

183

and geomagnetic field models.

184

3 METHOD FOR CHARACTERIZING THE TIME-SPECTRA OF GAUSS

185

COEFFICIENTS

186

Assuming that all Gauss coefficients but the axial dipole sample stochastic Gaussian processes of auto-

187

covariance function (13), we use numerical geodynamo simulations to discuss how to estimate the two

188

parameters σ and ω that characterize the processes. Then, we compare the theoretical power spectrum

189

of these processes to the actual spectrum of the Gauss coefficients in our numerical simulations.

190

3.1 Simulations used in the study

191

We rely on three dynamo numerical simulations named Step 0 (S0), Step 1 (S1), and Coupled Earth

192

(CE). All three solve the momentum, codensity and induction equations under the Boussinesq approx-

193

imation, for an electrically conducting fluid within a spherical shell of aspect ratio 0.35 between the

194

inner core and the outer core of radius c. S0 and S1 were computed using the free XSHELLS code

195

(Schaeffer 2015), assuming no-slip and fixed homogeneous heat flux conditions at both the inner and

196

outer boundaries. CE (Aubert et al. 2013) was run using the PARODY-JA code (Aubert et al. 2008),

197

assuming no-slip conditions at the inner boundary, free-slip conditions at the outer core boundary, and

198

heterogeneous mass-anomaly flux both at the inner and at the outer boundaries. This simulation also

199

includes a gravitational coupling between the inner core and the mantle. Both codes use finite dif-

200

ferences in radius and spherical harmonic expansion (Schaeffer 2013), together with a semi-implicit

201

Crank-Nicolson-Adams-Bashforth time scheme of order 2.

202

Non-dimensional parameters and times characterizing these simulations are given in Table 1. Di-

203

mensionless times are transformed into years following Lhuillier et al. (2011b) – see also sections 3.2

204

and 4.3. The field intensity is also rescaled to dimensional units using a proportionality constant such

205

that the averaged root mean square (r.m.s.) field in the shell is equal to 4 mT, a value comparable to

206

that estimated for the Earth’s core by Gillet et al. (2010).

207

The longest simulations S0 and CE allow to investigate long time-scales, whereas the high sam-

208

pling rate and the small Ekman number in S1 give access to shorter time-scales. All three simulations

209

are dipole-dominated at the Core Mantle Boundary (CMB); see the relative dipole field strength f dip

210

in Table 1. They also display non-dipolar structures and significant secular variation (but no polarity

211

reversal). The field in CE has the particularity to show prominent equatorial structures that undergo

212

a westward drift, as observed for the Earth’s magnetic field over the past four centuries (Finlay and

213

Jackson 2003). It is also important to notice that the magnetic Reynolds number Rm (defined as the

214

ratio of magnetic diffusion time over advection time) in our three simulations is close to the Earth’s

215

core value (see Table 1).

216

Statistics over periods much longer than a few 10,000 years (e.g. involving reversals) would re-

217

quire much longer simulations. There is thus a trade-off between capturing the long term evolution

218

of dipole moment changes and reproducing rapid field variations (Meduri and Wicht 2016). Robust

219

estimates of the mean dipole field strength require simulations over many diffusion times that are

220

presently achievable only for large Ekman number (e.g., Olson et al. 2012; Davies and Constable

221

2014). Because we are particularly interested here into decadal to millennial time-scales, we use pa-

222

rameters closer to (yet still far away from) the geophysical ones. Our simulations thus cover a wide

223

range of periods shorter than the turn-over time t _U .

224

We show in Fig. 1 and 2 examples of the time series that we analyse in the rest of the paper.

225

The axial dipole has a non-zero mean value and displays large long-period fluctuations. We observe a

226

decrease of both the amplitude and the time-scale of fluctuations of the other coefficients with degree.

227

While temporal fluctuations of all coefficients seem rather stationary in simulations S0 and CE (Fig.

228

1), non-stationarity is observed in the shorter simulation S1 for the largest degrees (Fig. 2, right). Note

229

that periodic oscillations are observed for coefficient G 2 ¹ in CE. These oscillations will be discussed in

230

section 4.2.

231

3.2 Variance and correlation time of Gauss coefficients

232

The magnetic field B outside the core is described through a scalar potential V such that B = −∇ V .

233

In this work, Gauss coefficients G n ^m and H ^m n are defined at the core surface (and not at the Earth’s

234

surface) with n and m the spherical harmonics degree and order, N the truncation degree, hence V is

235

decomposed as

236

V (r, θ, φ, t) = c

N

X

n=1

c r

n+1 n

X

m=0

( G n ^m (t) cos mφ + H ^m n (t) sin mφ) P _n ^m (cos θ) , (14)

237

Name Definition S0 S1 CE C-600 C-1400 Earth’s core

Ekman E = ν/(ΩD ² ) 10 ^{− 5} 10 ^{− 6} 3 10 ^{− 5} 5 10 ^{− 5} 5 10 ^{− 5} 4 10 ^{− 15}

Flux Rayleigh Ra F = F D ² /4πρκ ² ν 4.4 10 ¹⁰ 8.9 10 ¹¹ 1.0 10 ⁹ 3.1 10 ⁷ 1.5 10 ⁸ ?

Magnetic Reynolds Rm = U D/η 710 660 940 42 90 1700

Prandtl P r = ν/κ 1 1 1 1 1 0.1 − 10

Magnetic Prandtl P m = ν/η 0.4 0.2 2.5 0.5 0.5 2 10 ^{− 6}

Alfv´en time t A = D √ µ 0 ρ/B 100 47 110 83 2

Dipole decay time t d = c ² /(π ² η) 1.2 10 ⁴ 1.2 10 ⁴ 3.2 10 ⁴ 1.2 10 ³ 2.7 10 ³ 5 10 ⁴

Turn-over time t U = D/U 69 76 140 120 120 120

Dissipation time ^† τ _diss ^mag 12 14.5 41

Dipole field strength ^‡ f dip 0.73 0.68 0.75 0.68

Simulation duration 85. 10 ³ 7.6 10 ³ 84. 10 ³ 91. 10 ³

Sampling interval 38 0.25 5.3 11

Table 1. Non-dimensional numbers and time-scales for numerical simulations and the Earth’s core. All times are given in years. D is the shell thickness, c is the outer core radius, B and U the root mean square of the magnetic field intensity and of the velocity in the fluid shell, Ω the rotation rate, η the magnetic diffusivity, ν the kinematic viscosity, κ the thermal diffusivity, µ 0 the magnetic permeability of free space, F the mass anomaly flux at the Inner-Core boundary (chemical convection, see Aubert et al. 2013). C-600 and C-1400 stand for the Calypso simulations of Buffett and Matsui (2015), after translating their time-scale into the τ SV - based scaling used throughout this paper, with τ SV = 14 t d /Rm (Lhuillier et al. 2011a). See Backus et al.

(1996, pp 200-204) for the calculation of the dipole decay time t d . ^† We refer to Christensen and Tilgner (2004) for the definition of the magnetic dissipation time τ _diss ^mag , ratio of magnetic energy to Ohmic dissipation. ^‡ The relative dipole field strength at the core surface f dip is defined as in Christensen and Aubert (2006). We have adopted ν = 1.5 10 ^{− 6} m ² s ^{− 1} , η = 0.75 m ² s ^{− 1} , ρ = 1.1 10 ⁴ kg.m ^{− 3} , τ SV = 415 yrs and B = 4 10 ^{− 3} T to give values for the Earth’s core. The turn-over time deduced from τ SV and Lhuillier et al. (2011a), t U = D/U ' 0.3 τ SV ' 125 yrs, is consistent with U ∼ 20 km.yr ^{− 1} in the Earth’s core and is within a factor of two of our estimates from simulations.

where r is the distance to the Earth center, θ the colatitude, φ the longitude, and P _n ^m are the Schmidt

238

quasi-normalized Legendre functions. We define the spatial power spectra for the geomagnetic field

239

and its secular variation

240



 

 



 

 



R _n = (n + 1)

n

X

m=0

h E( G n ^m 2 ) + E( H ^m n 2 ) ⁱ S _n = (n + 1)

n

X

m=0

h E(∂ _t G n ^m 2 ) + E(∂ _t H ^m n 2 ) ⁱ

(15)

0 10 20 30 40 50 60 70 80

−2000

−1500

−1000

−500 0 500

Times (bottom: kyr, top: τ_SV)

Gauss coefficient (µT)

g₁⁰ g₂¹ g₅¹

0 50 100 150 200

0 10 20 30 40 50 60 70 80

−400

−350

−300

−250

−200

−150

−100

−50 0 50 100

Times (bottom: kyr, top: τ_SV)

Gauss coefficient (µT)

g1

0 g

2

1 g

5 1

0 50 100 150 200

Figure 1. Time series of coefficients G 1 ⁰ , G 2 ¹ and G 5 ¹ from simulations S0 (left) and CE (right). The top scale gives the dimensionless time (based on τ SV ).

as functions of degree n, from which a correlation time τ _n = ^p R _n /S _n can be derived (Hulot and

241

Le Mou¨el 1994).

242

The two quantities R n and τ n are now assumed to follow simple laws as a function of the degree

243

n (for n ≥ 2):

244

R _n ' αβ ⁿ , τ _n ' δn ^−γ . (16)

245

Constable and Parker (1988b) found that geomagnetic field models (1 ≤ n ≤ 12) are consistent with

246

β = 1, whereas Roberts et al. (2003) inferred β ' 0.90 from observations for n ≥ 3. Holme and

247

Olsen (2006) and Lesur et al. (2008) examined their satellite field models and estimated γ ' 1.45 and

248

γ ' 1.375 respectively whereas Christensen and Tilgner (2004) and Lhuillier et al. (2011b) argued

249

80 81 82 83 84

−8000

−6000

−4000

−2000 0 2000 4000 6000 8000 10000

Times (bottom: kyr, top: τ_SV)

Gauss coefficient (µT)

g₂¹ g₅¹ g₁₂¹

192 194 196 198 200 202

3 4 5 6 7

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

x 10

Times (bottom: kyr, top: τ

)

Gauss coefficient ( µ T)

g

1

g

5

1

g

12 1

8 10 12 14 16 18

Figure 2. 5 kyr time series of coefficients G 2 ¹ , G 5 ¹ and G 12 ¹ from simulations CE (left) and S1 (right). The top

scale gives the dimensionless time (based on τ SV ).

instead for γ = 1 in joint analyses of geodynamo simulations and geomagnetic field models. The

250

latter authors also scaled time in simulations so that τ _SV = δ | γ=1 matches the geophysical value and

251

estimated τ _SV = 415 years from a fit of τ n for degrees n ∈ [2 − 13].

252

Building on these works, we shall assume β = γ = 1 hence a flat spatial power spectrum R n at the

253

CMB for the observable length-scales. This simplification allows to easily convert numerical times into

254

years. The remaining parameters (α, δ) entering equations (16) can be derived from the average of R _n

255

and a least-squares fit of log(τ n ) versus log(n). Since these two quantities are not normally distributed,

256

a more accurate estimate may be obtained using a maximum likelihood approach, as developed by

257

Lhuillier et al. (2011b) for τ _n (see appendix A). We discuss in Appendix B the estimation of the

258

parameters of the regression model (16) as the conditions β = γ = 1 are relaxed.

259

For each simulation, we have computed different estimates of the spatial power spectrum R n and

260

of the time τ n : an ensemble of instantaneous values ( R ˆ n , ˆ τ n ) averaged over m (0 ≤ m ≤ n) only, an

261

estimate (R _n , τ _n ) averaged over m and the total duration of the simulations, and the similarly averaged

262

(R ^∗ _n , τ _n ^∗ ) once subtracted the mean values of the coefficients. Time-averaged estimates (R _n , τ _n × n)

263

and (R ^∗ _n , τ _n ^∗ × n) are shown in Fig. 3 for the three simulations. We also represent the fits R n = α

264

and τ n × n = δ calculated either with the least-square method or the maximum likelihood one.

265

In addition, we plot two-sigma intervals for α and δ deduced from an ensemble of ten snapshots.

266

Overall, the different time-averaged estimates of α and δ yield rather similar results given the large

267

variability within the ensemble of snapshot estimates. Removing or not the average appears therefore

268

as a secondary issue.

269

Spectra R _n for CE and S0 simulations are almost flat, validating the hypothesis β = 1, while that

270

for the most extreme (lowest viscosity, strongest forcing) simulation S1 presents a slightly decreas-

271

ing trend with n, closer to current estimates from geomagnetic field models, as further discussed in

272

Appendix B.

273

Times τ _n reflect slightly different behaviors in all three simulations. If the hypothesis γ = 1

274

agrees well with the outputs from CE, S1 (resp. S0) favors instead a slightly larger (resp. lower)

275

exponent. In simulation S1, we obtain a γ value closer to 1 after removing the time-average value of

276

the coefficients, which mainly affects τ _n estimates at low degrees. Furthermore, we note a wide time

277

variability in the instantaneous estimates τ ˆ _n , suggesting that a snapshot estimate alone, as available

278

from modern geophysical observations (see e.g. Holme et al. 2011) for which the long-term average

279

of coefficients is not available, is insufficient to determine precisely γ. All in all, we conclude that the

280

simple hypothesis γ = 1 is consistent with our three simulations (see Appendix B for more details). An

281

error of the order of 50% may occur when measuring the magnitudes of α and δ from instantaneous

282

values, as shown by the two-sigma interval in Fig. 3 (right) and in table A2. This translates into a

283

2 4 6 8 10 12 1

1.5 2 2.5

3 x 10 ¹¹

Degree n R n (nT 2 )

2 4 6 8 10 12

200 300 400 500 600 700

Degree n τ n × n (left: yr, right: τ S V )

0.6 0.8 1 1.2 1.4 1.6

2 4 6 8 10 12

0 0.5 1 1.5

2 x 10 ¹⁰

Degree n R n (nT 2 )

2 4 6 8 10 12

300 400 500 600

Degree n τ n × n (left: yr, right: τ S V )

0.8 1 1.2 1.4

2 4 6 8 10 12

1 1.5 2 2.5 3

3.5 x 10 ¹⁰

Degree n R n (nT 2 )

2 4 6 8 10 12

300 400 500 600 700 800

Degree n τ n × n (left: yr, right: τ S V )

0.8 1 1.2 1.4 1.6 1.8

Figure 3. Spatial power spectrum R n (left) and n × τ n (right) as a function of the spherical harmonic degree n for simulations S0 (top), CE (middle) and S1 (bottom) from the expected variances as in equation (15), either removing (blue dots) or keeping (red dots) the time-average of the coefficients. Dashed (resp. solid) colored lines stand for estimates of α and δ using least-squares (resp. maximum likelihood) regressions (16) with β = γ = 1.

Grey lines represent the two-sigma intervals around the average of 10 estimates of α and δ from independent

snapshots R ˆ n and τ ˆ n , which are not represented. The right scale on the n × τ n plots gives the dimensionless

time in τ SV units.

variability in τ SV significantly larger than that observed by Lhuillier et al. (2011b) from a dynamo

284

simulation at larger viscosity and lower forcing.

285

Note that the time-series of non-dipole coefficients represented on Fig. 2 appear uncorrelated when

286

sampled over periods longer than 2πτ n = 2πτ SV /n (i.e. for periods longer than about 1300 yr, 500

287

yr, and 200 yr for degrees 2, 5, and 12 respectively). This suggests a flat power spectrum at lower

288

frequencies, as expected for the two-parameter AR2 processes described in section 2.2

289

3.3 Frequency spectra of Gauss coefficients

290

In order to avoid frequency leakage when estimating the power spectrum for the finite-length time-

291

series of Gauss coefficients, we adopt a multi taper approach (e.g., Percival and Walden 1993). The

292

advantage of this approach is that the power spectrum variance is reduced by averaging independent

293

estimates of the power spectrum obtained after multiplying the series by various orthogonal tapers.

294

Several variants of the multi taper approach have been used before to assess the power spectrum of

295

the dipole moment. Constable and Johnson (2005) relied on sine tapers (Riedel and Sidorenko 1995).

296

Olson et al. (2012) chose instead to break the series into overlapping segments tapered using a Hanning

297

window (Welch 1967). As Buffett and Matsui (2015), we adopt in this study an approach based on

298

Slepian functions (Thomson 1982). We use seven Slepian tapers characterized by a power spectrum

299

with energy concentrated in a bandwidth [ − W, W ], where W = 4/(N ∆t), N is the number of data,

300

and ∆t is the sampling interval. As a consequence, the power spectrum estimated at a given frequency

301

f is controlled by values of the power spectrum within [f − W, f + W ], with W the resolution of the

302

power spectrum.

303

We test the multi taper approach of Thomson (1982) on a realisation of a stochastic process.

304

The obtained spectra are further smoothed by running averages over a length that linearly increases

305

with the frequency (from 1 point at minimum frequency to 201 points at maximum frequency). We

306

show the spectra obtained for this realisation both before and after removing its averaged value (Fig.

307

4). Although these power spectra include a certain amount of noise, they reproduce well both the

308

amplitude and the spectral indices of the true power spectrum, except at frequencies lower than the

309

resolution W . At frequencies f < W , the average value of the series influences the power spectra,

310

which strongly differ whether the average is removed or not: the spectrum obtained without removing

311

the average shows a step at low frequencies, which is an artefact. The above method for calculating

312

spectra is used below for all our results. Note that we do not remove linear trends in the time-series

313

before computing the spectra. Nevertheless, we checked that the shape of the spectra computed here

314

with the multi taper approach is not significantly different whether the trend has been removed or not.

315

Fig. 5 displays power spectra for degree 5 Gauss coefficient time series at the CMB, from the three

316

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

10¹ 10³ 10⁵ 10⁷

Frequency (yr⁻¹) Power spectrum (µT2.yr)

Figure 4. Comparison of power spectra for random time-series, estimated using the multi taper approach applied before (blue) and after (red) removing the averaged value of the series, and superimposed on the theoretical power spectrum (black). The series parameters are chosen to mimic a plausible behavior for the axial dipole coefficient (at the Earth’s surface): it is a Gaussian random series with an averaged value of -35 µT, a standard deviation of 5 µT, with a two-parameters AR2 auto-covariance function as defined in equation (13), with ω ^{− 1} = 500 yrs. The theoretical power spectra of this series is given in equation (12). The series contains N = 2000 data sampled every ∆t = 50 years. The vertical black line indicates the value of the concentration half-bandwidth W = 4/(N ∆t) of the Slepian tapers. These spectra were obtained using the subroutine pmtm from Matlab

and then further smoothed using running averages.

simulations. For the two longest simulations (S0 and CE), we observe that spectra for all coefficients

317

are flat (or white) at low frequencies, and show a constant spectral index at high frequencies, hinting

318

to a scale invariance. The change of spectral index occurs within a narrow band of frequencies, and

319

the cut-off frequency between the two regions of the spectra increases with the spherical harmonic

320

degree, as illustrated in Fig. 6 for the CE dynamo. Whereas the spectral index at large frequencies

321

appears independent of the spherical harmonic order in S0, it significantly increases with m in the CE

322

and S1 simulations. Power spectra obtained from S1 do not show a flat plateau at low frequencies as

323

a consequence of the short duration of the simulation: we do not have access to long enough periods

324

to reach the domain where P ∝ f ⁰ . Spectra for this simulation show a steep decrease with f at high

325

frequency, which is absent in the S0 and CE spectra.

326

3.4 Comparison with the spectrum of a two-parameter AR2 process

327

Expression (13) corresponds to a particular autoregressive process of order 2 that only depends on two

328

parameters, a variance σ ² and a characteristic time-scale ω ⁻¹ . As in Gillet et al. (2013), we further

329

assume that these two parameters only depend on the spherical harmonic degree n, which amounts

330

to posit that the statistics of the field are independent of longitude and latitude (Hulot and Bouligand

331

10

10

10

10

10

10

10

10

frequency (bottom: yr

, top: τ

) power spectrum (nT

.yr)

10

10

10

S0, n=5

10

10

10

10

10

10

10

10

10

10

10

frequency (bottom: yr

, top: τ

) power spectrum (nT

.yr)

10

10

10

10

CE, n=5

10

10

10

10

10

10

10

10

10

10

10

10

10

frequency (bottom: yr

, top: τ

) power spectrum (nT

.yr)

10

10

10

10

S1, n=5

Figure 5. Power spectra computed using the multi taper approach of Thomson (1982) for coefficients of degree

n = 5, from simulations S0 (top), CE (middle), and S1 (bottom). All coefficients G n ^m and H ^m n of order m ∈

[0, n] are represented with gradually varying colors (from red for m = 0 to blue for m = n). The black

solid (dashed) curves display the power spectra (12) with parameters τ n and σ n estimated using the maximum

likelihood method and using time averaged Gauss coefficients variances in (15), once removed (or not) their

averaged value. The grey lines represent the two-sigma interval around the average of 10 power spectra with

parameters ω _n ^{− 1} and σ n deduced from independent snapshot R ˆ n and τ ˆ n . The thin vertical black line indicates

the resolution half-bandwidth. The top scale gives the dimensionless frequency (based on τ _SV ^{− 1} ).

2005). Then, for each degree n, one deduces from (15) that σ _n ² = R n /(n + 1)(2n + 1), and from

332

equations (5) and (6) the relation ω _n ⁻¹ = τ _n ; these two parameters define the auto-covariance functions

333

C n (τ ).

334

Since long enough geophysical series to produce statistical averages are not available, Gillet et al.

335

(2013) approximated (R _n , τ _n ) by the quantity ( R ˆ _n , τ ˆ _n ) estimated from a snapshot of the well doc-

336

umented (and supposedly representative) satellite era. This approximation relies on the assumption

337

that main field and secular variation series are unbiased, i.e. E( G n ^m ) = E( H ^m n ) = E(∂ t G n ^m ) =

338

E(∂ _t H ^m n ) = 0. This assumption is certainly not valid for the axial dipole between two polarity rever-

339

sals. For this reason, Hellio et al. (2014) considered instead dipole deviations in the expression (15)

340

for n = 1. We test here the validity of using snapshot estimate ( R ˆ n , τ ˆ n ) to define the auto-covariance

341

function of non-dipole coefficients.

342

For each simulation, we estimate parameters α and δ entering (16) (with β = γ = 1) using

343

both averaged and instantaneous estimates of the spatial power spectrum and correlation times (i.e.,

344

( ¯ R n , ¯ τ n ), (R ^∗ _n , τ _n ^∗ ) and ( ˆ R n , τ ˆ n )) and a maximum likelihood approach. α and δ are then used to de-

345

termine variances σ _n ² and correlation times ω ⁻¹ _n , and to predict the theoretical spectrum (12) for all

346

degrees n. We then estimate a two-sigma interval from 10 spectra (12) deduced from snapshots. These

347

curves are superimposed in Fig. 5 (for n = 5) and Fig. 6 (CE simulation for n = 2, 5, 12) on spectra

348

of the Gauss coefficients.

349

For all three simulations and all degrees, we observe overall a good agreement between the differ-

350

ent theoretical spectra, with some discrepancies that we detail in the next paragraph. The theoretical

351

spectra obtained from averaged estimates once removed or not the coefficient averaged value are very

352

close, suggesting that the assumption of unbiased series is valid. The two-sigma intervals are relatively

353

narrow compared to the noise level in the individual spectra and to the variability among spectra of

354

same degree, showing that the use of snapshot estimates is appropriate.

355

For simulation S0, the power-spectra calculated from (12) reproduce very well the power spectra

356

of the field coefficients at all frequencies. For simulation CE, the spectrum (12) approximates relatively

357

well the power spectra of low order Gauss coefficients for all degrees n. On the other hand, the power

358

spectra for the largest order coefficients (m ∼ n) decreases more rapidly than f ⁻⁴ at its high frequency

359

end. Simulation S1 also presents, at periods shorter than 10 years, Gauss coefficient power spectra

360

steeper than f ⁻⁴ . Buffett and Matsui (2015) conjecture that the occurrence of a period range presenting

361

a s = 6 spectral index, as observed from numerical computations (Olson et al. 2012; Davies and

362

Constable 2014), could be related with a mechanism involving magnetic diffusion below the CMB.

363

However, the identification of a spectral index s requires a power-law behavior P(f) ∝ f ^−s over a

364

significant frequency range. Instead, a power spectrum P (f ) ∝ exp( − f), which is reminiscent of a

365

10

10

10

10

10

10

10

10

10

10

10

10

10

frequency (bottom: yr

, top: τ

) power spectrum (nT

.yr)

10

10

10

10

CE, n=2

10

10

10

10

10

10

10

10

10

10

10

frequency (bottom: yr

, top: τ

) power spectrum (nT

.yr)

10

10

10

10

CE, n=5

10

10

10

10

10

10

10

10

10

frequency (bottom: yr

, top: τ

−1

) power spectrum (nT

.yr)

10

10

10

10

CE, n=12

Figure 6. Power spectra for Gauss coefficients series of spherical harmonic degrees 2, 5 and 12, from the CE

simulation. Same colors and line types as in Fig. 5.

dissipation range (see e.g. Frisch 1995), may arguably be observed at high frequencies in simulation

366

S1. Hence, the narrow range of frequencies that displays a spectral index of 4 may result from too

367

important diffusive processes in simulations (see § 4.3).

368

4 DISCUSSION

369

4.1 Model for dipole fluctuations

370

The minimal model (12), which appears appropriate for all Gauss coefficients but the axial dipole in

371

our simulations, involves only one time-scale ω ⁻¹ . It can be presented (see § 2.1) as a special case

372

(ω = χ, i.e. τ _s = τ _f ) within a more general family of models (3) having two distinct time-scales ω ⁻¹

373

and χ ⁻¹ – or equivalently τ _s and τ _f , see equation (11). For ω < χ, the associated power spectra (3)

374

show a power law in f ⁻² at intermediate frequencies – between frequencies 1/(2πτ s ) and 1/(2πτ _f ).

375

For this reason, they were employed by Buffett and Matsui (2015) to account for the spectrum of the

376

axial dipole as inferred from numerical simulations and from geomagnetic models. We concur with

377

these results. In the two simulations S0 and CE that are long enough to address long-lived dipole

378

fluctuations, the power spectrum for the axial dipole coefficient G 1 ⁰ does not present a sharp transition

379

from 0 to 4 spectral index (see Fig. 7). Contrary to the equatorial dipole coefficients G 1 ¹ and H ¹ 1 ,

380

whose spectra are well fitted by a two parameters AR2 spectrum (12), the spectrum for G 1 ⁰ shows an

381

intermediate spectral index over about one decade, which is well fitted by the three parameter function

382

(3).

383

The calculation of τ s and τ _f by Buffett and Matsui hinges on the determination of the two transi-

384

tion frequencies between domains of spectral index 4, 2, and 0 respectively (see § 2.1). Fig. 7 illustrates

385

our fit between the spectra for S0 and CE and the function (3) where we have entered our estimations

386

for ω and χ (directly related to τ _s and τ _f ). Table 2 gives a comparison between our results and the

387

values of τ s and τ _f calculated by Buffett and Matsui but scaled in units of τ SV . In S0 and CE, the

388

transition frequency between domains of spectral index s ' 2 and s ' 4 (Fig. 7) leads to τ f ' 65

389

and 125 yrs respectively, values about 2 to 3 times larger than the estimates by Buffett and Matsui.

390

Switching to long periods, they made the analogy between the times τ _s and t _d found in their simula-

391

tions. Although this analogy cannot be ruled out by our results, simulations S0 and CE show values of

392

the ratio t d /τ s significantly different from 1 (see Tables 1 and 2).

393

Unfortunately, the frequency range with a flat power spectrum is clear neither in the simulations

394

investigated here, nor in those of Buffett and Matsui. In both studies, this part of the power spectrum

395

is within the concentration bandwidth of the taper (see their Fig. 4 and our Fig. 7); we thus cannot

396

determine if this is to be associated with a real feature of the axial dipole power spectrum, or with

397

10

10

10

10

10

10

10

10

10

10

frequency (bottom: yr

, top: τ

) power spectrum (nT

.yr)

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

frequency (bottom: yr

, top: τ

) power spectrum (nT

.yr)

10

10

10

10

Figure 7. Power spectra (red) for the axial dipole series from simulations S0 (left), and CE (right). In black are superimposed the three parameters AR2 spectra (3) fitted to the series spectra for f > W – range covered by the blue fit. The parameter W denotes the resolution half-bandwidth. The axial dipole variance is obtained directly from the series (removing the average). The frequency ω is estimated from the square root of the ratio of the variances of G 1 ⁰ and ∂ G 1 ⁰ /∂t. The remaining parameter χ is obtained by minimizing the L2 norm of the difference between the logarithms of G 1 ⁰ series spectrum and of (3). The top scale gives the dimensionless frequency (based on τ _SV ^{− 1} ). The thin vertical line indicates f = W . Black segments indicate spectral indices of 2 and 4.

an artefact due to tapering. As a result, the estimates of τ s obtained from numerical simulations and

398

given in Table 2 are not very accurate. Nevertheless, all estimates for ω ⁻¹ = (τ s τ f ) ^1/2 obtained from

399

numerical series of the axial dipole are within a factor of 2 of the value that we would obtain by

400

extrapolating the relation ω ⁻¹ _n = τ _n = τ _SV /n (used for non-dipole coefficients) to the degree n = 1

401

(i.e., ω ⁻¹ = 415 yr) .

402

The time ω ⁻¹ inferred from paleo- and archeomagnetic models appears significantly longer than

403

estimates deduced from numerical simulations. In our opinion, the spectra of archeomagnetic field

404

models, in the high frequency range where the spectral index is s ' 4, are much influenced by the

405

regularization used in their construction. This explains why these models do not resolve geomagnetic

406

jerks.

407

4.2 Deviations from spherical symmetry

408

Whereas temporal spectra from simulation S0 are fairly independent of the order m for all degrees

409

but n = 1 (Fig. 5), suggesting that fluctuations of the non-dipole field are spherically symmetric at

410

the CMB, we detect some significant dependence on the order from computations CE and S1. In CE,

411

the spectra for coefficients of large order (m ' n) present a larger spectral index at high frequencies.

412

Model/Simulation τ s (yr) τ f (yr) ω ^{− 1} (yr) Reference

PADM2M ^{† 1} - CALS10k.1b ^{† 2} 29 000 100-200 1700-2400 Buffett et al. (2013); Buffett and Matsui (2015) Calypso (Rm=90) 1050 37 200 Buffett et al. (2014); Buffett and Matsui (2015)

Calypso (Rm=42) 1100 35 200 Buffett and Matsui (2015)

S0 3610 65 480

CE 3490 125 660

Table 2. Time-scales τ s and τ f involved to reproduce the power spectrum of the axial dipole deduced from archeo- and paleo-magnetic observations and from dynamo numerical simulations (see the definitions of τ s

and τ f in equations (8) and (9) respectively). The time ω ^{− 1} is obtained as (τ s τ f ) ^1/2 . The different times of the Calypso simulations have been converted into the τ SV -based scaling adopted throughout the paper, using t d = Rm × τ SV /14 (Lhuillier et al. 2011a). ^{† 1} Ziegler et al. (2011), ^{† 2} Korte and Constable (2011).

As a consequence, more energy is contained in coefficients of small order at high frequencies and in

413

coefficients of large order at intermediate frequencies (for periods typically from 100 to 1000 yrs).

414

Because spherical harmonics of low and large orders have their largest contributions at respectively

415

high and low latitudes, this suggests fluctuations at intermediate periods are stronger at low latitude

416

(equatorial features primarily project into sectorial coefficients). This likely reflects the westward drift

417

of low latitude structures observed in the CE simulation (see Aubert et al. 2013).

418

The power spectra for coefficients G 2 ¹ and H ¹ 2 in simulation CE (and to a lesser extent for order

419

1, degrees 4 and 6 coefficients, not shown) display a significant peak at periods around 2500 yrs (see

420

Fig. 6), which translates into quasi-periodic oscillations in the time-series (see Fig. 1, right). This

421

particular period corresponds to the time needed to circumnavigate the outer core at the average speed

422

of the westward drift (Aubert et al. 2013). These periodic variations mainly affect m = 1 coefficients

423

of the magnetic field through the advection of the eccentric gyre resulting, in the CE scenario, from

424

the heterogeneous heat fluxes.

425

The topology of field patches at the CMB is influenced by the underlying dynamics. Indeed, the

426

predominant Coriolis force in geodynamo simulations favors columnar structures aligned with the

427

rotation axis, and together with magnetic forces it textures the vorticity field in the equatorial plane

428

(e.g. Kageyama et al. 2008). As a result of field concentration by the vortices, the magnetic field at the

429

CMB (outside the polar caps above and below the inner core) shows thin filaments primarily aligned

430

along meridians (e.g. Takahashi and Shimizu 2012). This is illustrated in Fig. 8 for our lowest viscosity

431

case, the strongly forced computation S1. We have thus some evidence that the Gauss coefficients at

432

the core surface cannot be treated as independent variables.

433

We deduce the following consequences for the inversion of geomagnetic data. First, using an

434

AR2 autocorrelation function that is independent of the coefficient order as prior information for the

435

Figure 8. Full resolution snapshot of the radial magnetic field at the CMB for the S1 simulation, shown using an Aitoff projection. In this snapshot, the maximum intensity of the magnetic field at the CMB is about 7 mT.

inversion of geomagnetic models may penalize actual features of the geomagnetic field such as the

436

westward drift of equatorial flux patches (Finlay and Jackson 2003) or periodic signals. Second, ac-

437

counting for spatial cross-covariances (as performed with twin experiments on geodynamo simulations

438

by Fournier et al. 2013) may improve the construction of prior information in field modeling studies.

439

4.3 Mechanisms underlying the different time-scales

440

Our approximation for the spectra of all coefficients but the axial dipole involves only one time-

441

scale ω ⁻¹ _n (= τ _SV /n). Lhuillier et al. (2011a) argued that τ _SV is related to the advection time t _U ,

442

τ _SV ' 3t _U ' 14t _d /Rm (see table 1 for definitions) and this relationship holds within a factor of 2

443

in our simulations. This link between τ SV and t U suggests that the advection time, or eddy turnover

444

time, controls the times ω ⁻¹ _n .

445

Our observation, from simulations S0 and CE, of a sharp transition between 0 and 4 spectral

446

index ranges suggests that fluctuations of non-dipole coefficients are controlled by a single time-scale,

447

or by two time-scales that are not significantly different. In our simulations, the axial dipole is the

448

only coefficient for which we found necessary to consider AR2 processes defined with two distinct

449

time-scales in order to account for the existence of a frequency range displaying a spectral index

450

of 2. One could wonder as Buffett et al. (2013) whether this is to be related to the specificity of

451

the axial dipole to show a non-zero average value. However, in this regard, our simulations may not

452